Contributions to the convolution and $\Psi$DO's over ultradistribution spaces

The convolution of distributions was studied from the early beggining of the distribution theory, by many authors. Important contribution was given by Professor Vladimirov. I have studied, with my students, convolution in various spaces of distributions and ultradistributions. The aim of this talk is to show that one can extend the Anti-Wick calculus over $\mathcal D^{\{M_p\}}(\mathbb{R}^d)$ for ultradistributions in ${\mathcal S'}_{\{A_p\}}^{\{M_p\}}(\mathbb{R}^d)$ with very weak assumptions on $A_p$ and conditions on $M_p$ related to the sequence $p!^m, m>1$ noted in the abstract. This is done by the use of the Wigner transform $W(\varphi,\varphi)$ with $\varphi $ being ultradifferentiable functions with the fast decrease as $|x|\rightarrow \infty.$ We develop the theory for $\varphi=e^{-r{\langle \cdot\rangle^q}},\; r>0, q\geq 1,$ as well as for $\varphi$ satisfying even faster decay. Special example is $\varphi= \exp{(-se^{{\langle\cdot\rangle}^q})}, s>0,q\geq 1.$ Note that we have given earlier a complete answer in our analysis related to the convolution with the kernel $e^{a|\cdot|^q},a>0$ and the related Anti-Wick calculus, in the case when $\varphi$ is a Gaussian.